The number of permutation equivalence classes of doubly-even binary linear codes is shown in the table below (counting only codes without zero columns), and the list of codes is linked from the list entries. For the smaller cases, you can load them directly in Sage:

Let F denote the field containing two elements. A linear binary code of type [N, k] is a vector subspace C of F^N with dimension k. Using the standard basis, the weight wt(v) of a vector v in an F-vector space is simply the number of coordinates equal to one. Define a doubly-even code to be a linear binary code with the constraint that every vector has weight divisible by 4. We use Gaborit's formulas for the number of distinct doubly-even codes.

An interesting fact is that doubly-even codes are guaranteed to be self-orthogonal, under the standard inner product . We can express the dot product of two vectors, modulo 2, as the number of coordinates where both vectors are one. We can express this in terms of weight: